Cluster sets and approximation properties of quasi-continuous and cliquish functions.

Abstract

The crucial concept for studying quasi-continuous and cliquish functions on arbitrary topological spaces $X$ is the concept of a semi-open subset of $X$. On the one hand, it gives rise to the cluster set $SO-C(f;x)$ of a function $f:X\rightarrow {\mathbb R}$ at a point $x\in X$, which turns out to be an appropriate tool for investigating both local and global properties of $f$. On the other hand, the concept of a semi-open set is used for introducing so-called semi-open partitions of $X$. A central result of the paper says that every quasi-continuous function can be represented as a uniform limit of step functions defined on a chain of semi-open partitions of $X$. Similarly, every cliquish function is proved to be the uniform limit of step functions defined on a chain of so-called almost semi-open partitions of $X$.